6.853: Topics in Algorithmic Game Theory - PowerPoint Presentation

Slide1

6.853: Topics in Algorithmic Game Theory

Fall 2011

Constantinos Daskalakis

Lecture 22Slide2

Characterizations of Incentive Compatible MechanismsSlide3

Characterizations

Only look at incentive compatible mechanisms (revelation principle)

When is a mechanism incentive compatible? Characterizations of incentive compatible mechanisms.

Maximization of social welfare can be implemented (VCG). Any others?

Basic characterization of implementable social choice functions.

What social choice functions can be implemented?Slide4

Direct CharacterizationSlide5

Direct Characterization

A mechanism is incentive compatible iff it satisfies the following conditions for every

i and every :

i.e., for fixed , there is an advertised price per alternative ; the bidder is free to affect the chosen alternative and through that the corresponding price that she’ll pay;

(ii)

i.e., for every , we have alternative where the quantification is over all alternatives in the range of

(

i

)

p

i

depends on only through the alternative Slide6

Direct Characterization (cont’d)

Proof

(if part): Denote ,

where respectively is the bidder’s true value and is a potential lie. Since the mechanism optimizes for

i

, the utility he receives when telling the truth is not less than the utility he receives when lying.

Slide7

Direct Characterization (cont’d)

Proof (cont):

(only if part; (

i

)) Suppose that for some ,

but . WLOG, assume that

. Then a player with type

will increase his utility by declaring .

(only if part; (ii)): Suppose and let instead

s.t

.

Now a player with type will increase his utility by

declaring .



Slide8

Weak MonotonicitySlide9

Weak Monotonicity

The direct characterization involves both the social choice function and the payment functions.

Weak

Monotonicity

provides a partial characterization that only involves the social choice function.Slide10

Weak Monotonicity

(WMON)

Def:

A social choice function satisfies Weak

Monotonicity

(WMON) if for all

i

, all we have that

i.e. WMON means that if the social choice changes when a single player changes his valuation, then it must be because the player increased his value of the new choice relative to his value of the old choice.Slide11

Weak Monotonicity

Theorem:

If a mechanism is incentive compatible then satisfies

WMON

. If all domains of preferences are convex sets (as subsets of an Euclidean space) then for every social choice function that satisfies

WMON

there exists payment function such that

is incentive compatible.

Remarks: (

i

) We will prove the first part of the theorem. The second part is quite involved, and will not be given here.

(ii) It is known that WMON is not a sufficient condition for incentive compatibility in general non-convex domains.Slide12

Weak Monotonicity

(cont’d)

Proof:

(First part) Assume first that is incentive compatible, and fix

i

and in an arbitrary manner. The direct characterization implies the existence of fixed prices for all

(that do not depend on ) such that whenever the outcome is then

i

pays exactly .

Assume . Since the mechanism is incentive compatible, we have

Thus, we haveSlide13

Minimization of Social Welfare

We know maximization of social welfare function can be implemented.

How about minimization of social welfare function?

No! Because of WMON.Slide14

Minimization of Social Welfare

Assume there is a single good. WLOG, let . In this case, player 1 wins the good.

If we change to , such that . Then player 2 wins the good. Now we can apply the

WMON

.

The outcome changes when we change player 1’s value. But according to

WMON

, it should be the case that . But . Contradiction. Slide15

Weak Monotonicity

WMON is a good characterization of implementable social choice functions, but is a

local one (i.e. a collection of local conditions). Is there a

global

characterization of what functions can be implemented, e.g. maximization of social welfare, etc.?Slide16

Weighted VCGSlide17

Affine Maximizer

Def:

A social choice function is called an

affine

maximizer

if for some

subrange

, for some weights and for some outcome weights , for every , we have that

i.e. maximization only over

A’

weighted social welfare

+ bonus per alternativeSlide18

Payments for Affine Maximizer

Proposition:

Let be an

affine

maximizer

. Define for every

i

,

where is an arbitrary function that does not depend on . Then, is incentive compatible.

the appropriate generalization of the VCG payment ruleSlide19

Payments for Affine Maximizer

Proof:

First, we can assume

wlog

. The utility of player

i

if alternative is chosen is . By multiplying by this expression is maximized when

is maximized which is what happens when

i

reports truthfully.



Slide20

Roberts Theorem

Theorem [Roberts 79]:

If , is onto , for every

i

, and

is incentive compatible then is an affine

maximizer

.

Remark:

The restriction is crucial (as in Arrow’s theorem); for the case , there do exist incentive compatible mechanisms beyond VCG.

|

A

|=2



single-parameter domains

the valuation function of each bidder is described by 1 parameter;

i.e

, for every bidder

i

there exists subset

W

i



A

(winning set) and a parameter

r

i

such that the bidder receives value

r

i

for any outcome in

W

i

and 0 otherwise; e.g. single-item/multi-unit auctions, etc.Slide21

Bayesian MechanismsSlide22

Bayesian Mechanism Design

Def:

A Bayesian mechanism design environment consists of the following:

setup

mech

+ for every bidder a distribution is known; bidder

i

’s type is sampled from it

Def:

A Bayesian mechanism consists of the following:

The utility that bidder

i

receives if the players’ actions are

x

1

,…,

x

n

is :Slide23

Strategy and Equilibrium

Def:

A strategy of a player i is a function .

Def

:

A profile of strategies is a

Bayes Nash equilibrium

if for all

i

, all , and all we have that

expected utility of bidder

i

for using

s

i

,

where the expectation is computed with respect to the actions of the other bidders assuming they are using their strategies

expected utility if bidder

i

uses a different action

x

i

’; still

the expectation is computed with respect to the actions of the other bidders assuming they are using their strategiesSlide24

First Price Auction

Theorem: Suppose that we have a single item to auction to two bidders whose values are sampled independently from [0,1]. Then the strategies

s1(t1)=

t

1

/2 and

s2

(

t

2

)

=

t

2

/2 are a Bayes Nash equilibrium.

Remark:

In the above Bayes Nash equilibrium, social welfare is optimized, since the highest winner gets the item.

Proof:

On the board.

Expected Payoff?

E[ max(X/2, Y/2)], where X, Y are independent U[0,1] random variables.

=1/3

Expected Payoff of second price auction?

E[

min(X, Y)

], where X, Y are independent U[0,1] random variables.

=1/3 !Slide25

Revenue Equivalence Theorem

All single item auctions that allocate (in

Bayes Nash equilibrium) the item to the bidder with the highest value and in which losers pay 0 have identical expected revenue.

Given a social choice function we say that a Bayesian mechanism

implements

if for

some

Bayes Nash equilibrium we have that for all types :

Generally:

Revenue Equivalence Theorem:

For two

Bayesian

-Nash

implementations of the same social choice function

f

,

*

we have

the following: if for

some type

t

0

i

of

player i, the expected (over the types of the other players) payment of player

i is the same in the two mechanisms, then it is the same for every value of ti.

In particular, if for each player

i

there exists a type

t

0

i

where the two mechanisms have the same expected payment for player

i

, then the two mechanisms have the same expected payments from each player and their expected revenues are the same.Slide26

Revenue Equivalence Theorem (cont.)

All single item auctions

that allocate (in Bayes Nash equilibrium) the item to the bidder with the highest value and in which losers pay 0 have identical expected revenue.

So

unless we modify the social choice function

we won’t increase revenue.

E.g. increasing revenue in single-item auctions:

- two uniform [0,1] bidders

- run second price auction with reservation price ½ (i.e. give item to highest bidder above the reserve price (if any) and charge him the second highest bid or the reserve, whichever is higher.

Claim:

Reporting the truth is a

dominant strategy equilibrium

. The expected revenue of the mechanism is 5/12 (i.e. larger than 1/3).